Some further clarification on Markov chain random fields and transiograms

The Markov chain random field (MCRF) theory and the transiogram spatial measure were proposed several years ago. Basic sequential simulation algorithms based on simple MCRF models such as the Markov chain sequential simulation algorithm and the Markov chain sequential co-simulation algorithm have been developed and used in a series of application studies. However, misunderstanding of these two ideas and the geostatistical approach built on them arose recently among some researchers in geostatistics. The purpose of this article is to further clarify some issues related to these two ideas, so as to avoid further misunderstanding. For those issues already clarified, trivial, or obviously irrelevant, we do not talk about them here.